1. Ch4 Sinusoidal Steady State Analysis
Engineering Circuit Analysis
Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
4.2 Phasors
4.3 Phasor Relationships for R, L and C
4.4 Impedance
4.5 Parallel and Series Resonance
4.6 Examples for Sinusoidal Circuits Analysis
References: Hayt-Ch7; Gao-Ch3;

2. Ch4 Sinusoidal Steady State Analysis
Any steady state voltage or current in a linear circuit with a sinusoidal source is a sinusoid
All steady state voltages and currents have the same frequency as the source
In order to find a steady state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency)
We do not have to find this differential equation from the circuit, nor do we have to solve it
Instead, we use the concepts of phasors and complex impedances
Phasors and complex impedances convert problems involving differential equations into circuit analysis problems
 Focus on steady state; 􀂄 Focus on sinusoids.

3. Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Key Words:
Period: T ,
Frequency: f , Radian frequency 
Phase angle
Amplitude: Vm Im

4. Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
v、i t t1 t2
Both the polarity and magnitude of voltage are changing.

5. Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Period: T — Time necessary to go through one cycle. (s)
Frequency: f — Cycles per second. (Hz)
f = 1/T
Radian frequency(Angular frequency):  = 2f = 2/T (rad/s）
Amplitude: Vm Im
i = Imsint， v =Vmsint
v、i t  2
Vm、Im

6. Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Effective Roof Mean Square (RMS) Value of a Periodic Waveform — is equal to the value of the direct current which is flowing through an R-ohm resistor. It delivers the same average power to the resistor as the periodic current does.
Effective Value of a Periodic Waveform

7. Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Phase (angle)
Phase angle
0
①如果正弦波的起始最小值发生在时间起点之前，则为正值。
②如果正弦波的起始最小值发生在时间起点之后，则为负值。
<0

8. Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Phase difference
— v(t) leads i(t) by (1 - 2), or i(t) lags v(t) by (1 - 2)
— v(t) lags i(t) by (2 - 1), or i(t) leads v(t) by (2 - 1)
v、i t v i
Out of phase。 t
v、i v i
v、i t v i
In phase.

9. Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Review
The sinusoidal waves whose phases are compared must:
① Be written as sine waves or cosine waves.
② With positive amplitudes.
③ Have the same frequency.
360°—— does not change anything.
90° —— change between sin & cos.
180°—— change between + & -

10. Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Phase difference
P4.1,
Find If

11. Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Phase difference
P4.2,
v、i t v i
-/3
/3 • 
P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

12. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
A sinusoidal voltage/current at a given frequency , is characterized by only two parameters :amplitude an phase
Key Words:
Complex Numbers
Rotating Vector
Phasors

13. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
E.g. voltage response
Time domain
Frequency domain
Complex form:
Phasor form:
Angular frequency ω is known in the circuit.
A sinusoidal v/i
Complex transform
Phasor transform
By knowing angular frequency ω rads/s.

14. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Rotating Vector Im  t x y i t  Im i
P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？
t1
i(t1)
A complex coordinates number:
Real value:
Imag 14

15. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Rotating Vector Vm x y  
P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

16. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Numbers
— Rectangular Coordinates 
|A| a b
real axis
imaginary axis
— Polar Coordinates
P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？
j——旋转90的算子
conversion：

17. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Numbers
Arithmetic With Complex Numbers
Addition: A = a + jb, B = c + jd, A + B = (a + c) + j(b + d)
Real Axis
Imaginary Axis A B
A + B
P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

18. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Numbers
Arithmetic With Complex Numbers
Subtraction : A = a + jb, B = c + jd, A - B = (a - c) + j(b - d)
Real Axis
Imaginary Axis A B
A - B
P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

19. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Numbers
Arithmetic With Complex Numbers
Multiplication : A = Am  A, B = Bm  B
A  B = (Am  Bm)  (A + B)
Division: A = Am  A , B = Bm  B
A / B = (Am / Bm)  (A - B)
P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

20. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Phasors
A phasor is a complex number that represents the magnitude and phase of a sinusoid:
Phasor Diagrams
P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？
A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes).
A phasor diagram helps to visualize the relationships between currents and voltages.

21. Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Exponentials
A real-valued sinusoid is the real part of a complex exponential.
Complex exponentials make solving for AC steady state an algebraic problem.
P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

22. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Key Words:
I-V Relationship for R, L and C,
Power conversion

23. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Resistor
v~i relationship for a resistor
Suppose
Relationship between RMS:
v、i t v i
Wave and Phasor diagrams：

24. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Time domain frequency domain
Resistor
With a resistor θ﹦φ, v(t) and i(t) are in phase .

25. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Resistor
Power
Transient Power
p0
Note: I and V are RMS values.
v、i t v i
Average Power
P=IV

26. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Resistor
P4.4 ， , R=10，Find i and P。

27. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor
v~i relationship
Suppose

28. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor
v~i relationship
Relationship between RMS:
For DC，f = 0，XL = 0.
v(t) leads i(t) by 90º, or i(t) lags v(t) by 90º

29. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor
v ~ i relationship
i(t) = Im ejwt
Represent v(t) and i(t) as phasors:
The derivative in the relationship between v(t) and i(t) becomes a multiplication by j in the relationship between and
The time-domain differential equation has become the algebraic equation in the frequency-domain.
Phasors allow us to express current-voltage relationships for inductors and capacitors in a way such as we express the current-voltage relationship for a resistor.

30. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor
v ~ i relationship
Wave and Phasor diagrams：
v、i t v i eL

31. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor
Power P t
Energy stored:
v、i t v i + -
Average Power
Reactive Power
（Var）

32. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor
P4.5，L = 10mH，v = 100sint，Find iL when f = 50Hz and 50kHz.

33. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor
v ~ i relationship
Suppose:
Relationship between RMS:
For DC，f = 0， XC  
i(t) leads v(t) by 90º, or v(t) lags i(t) by 90º

34. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor
v ~ i relationship
v(t) = Vm ejt
Represent v(t) and i(t) as phasors:
The derivative in the relationship between v(t) and i(t) becomes a multiplication by j in the relationship between and
The time-domain differential equation has become the algebraic equation in the frequency-domain.
Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor.

35. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor
v ~ i relationship
Wave and Phasor diagrams：
v、i t v i

36. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor
Power P t
Energy stored:
v、i t v i + -
Average Power: P=0
Reactive Power
（Var）

37. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor
P4.7，Suppose C=20F，AC source v=100sint，Find XC and I for f = 50Hz, 50kHz。

38. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Review (v-I relationship)
Time domain
Frequency domain ,
, v and i are in phase.
, v leads i by 90°.
, v lags i by 90°. R C L

39. Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Summary R： L： C：
Frequency characteristics of an Ideal Inductor and Capacitor:
A capacitor is an open circuit to DC currents;
A Inducter is a short circuit to DC currents.

40. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Key Words:
complex currents and voltages.
Impedance
Phasor Diagrams

41. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex voltage， Complex current， Complex Impedance
AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm’s law:
Z is called impedance.
measured in ohms ()

42. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
Complex impedance describes the relationship between the voltage across an element (expressed as a phasor) and the current through the element (expressed as a phasor)
Impedance is a complex number and is not a phasor (why?).
Impedance depends on frequency

43. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
ZR = R  ＝ 0; or ZR = R  0
Resistor——The impedance is R or
Capacitor——The impedance is 1/jwC or
Inductor——The impedance is jwL

44. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
Impedance in series/parallel can be combined as resistors.
Voltage divider:
Current divider:

45. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
P4.8,

46. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
Phasors and complex impedance allow us to use Ohm’s law with complex numbers to compute current from voltage and voltage from current
20kW + -
1mF
10V  0 VC
w = 377
Find VC
P4.9
How do we find VC?
First compute impedances for resistor and capacitor:
ZR = 20kW = 20kW  0
ZC = 1/j (377 *1mF) = 2.65kW  -90

47. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
20kW + -
1mF
10V  0 VC
w = 377
Find VC
P4.9
20kW  0 + -
2.65kW  -90
10V  0 VC
Now use the voltage divider to find VC:

48. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
Impedance allows us to use the same solution techniques for AC steady state as we use for DC steady state.
All the analysis techniques we have learned for the linear circuits are applicable to compute phasors
KCL & KVL
node analysis / loop analysis
superposition
Thevenin equivalents / Norton equivalents
source exchange
The only difference is that now complex numbers are used.

49. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Kirchhoff’s Laws
KCL and KVL hold as well in phasor domain.
KCL:
ik- Transient current of the #k branch
KVL：
vk- Transient voltage of the #k branch

50. Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Admittance
I = YV, Y is called admittance, the reciprocal of impedance, measured in siemens (S)
Resistor:
The admittance is 1/R
Inductor:
The admittance is 1/jL
Capacitor:
The admittance is j  C

51. – Ch4 Sinusoidal Steady State Analysis 4.4 Impedance + Phasor Diagrams
A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes).
A phasor diagram helps to visualize the relationships between currents and voltages.
I = 2mA  40, VR = 2V  40
VC = 5.31V  -50, V = 5.67V  
2mA  40 –
1mF VC +
1kW VR V
Real Axis
Imaginary Axis VR VC V

52. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Key Words:
RLC Circuit,
Series Resonance
Parallel Resonance

53. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series RLC Circuit
(2nd Order RLC Circuit ) v
vR
vL
vC
Phasor 

54. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series RLC Circuit
(2nd Order RLC Circuit ) Z
X = XL-XC R 
Phase difference: 
XL>XC   >0，v leads i by ——Inductance Circuit
XL<XC   <0，v lags i by ——Capacitance Circuit
XL=XC   =0，v and i in phase——Resistors Circuit

55. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series RLC Circuit
(2nd Order RLC Circuit ) v
vR
vL
vC

56. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series RLC Circuit
(2nd Order RLC Circuit )
P4.9, R. L. C Series Circuit，R = 30，L = 127mH，C = 40F，Source
. Find 1) XL、XC、Z；2) and i; 3) and vR;
and vL; and vC; 4) Phasor diagrams; v
vR
vL
vC
P4.10，Computing by (complex numbers) Phasors

57. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance
(2nd Order RLC Circuit )
Resonance condition
Resonance frequency
and
——Series Resonance f0 f X

58. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance
(2nd Order RLC Circuit )
Resonance condition:
Zmin；when V=constant, I=Imax=I0。
When , 
Quality factor Q,

59. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance
(2nd Order RLC Circuit )

60. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance
(2nd Order RLC Circuit )

61. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance
(2nd Order RLC Circuit )

62. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance
(2nd Order RLC Circuit )

63. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance
(2nd Order RLC Circuit )

64. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance
(2nd Order RLC Circuit )

65. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Parallel RLC Circuit  
When ,
In phase with
Parallel Resonance
Parallel Resonance frequency
In generally
Zmax Imin:

66. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Parallel RLC Circuit  
Z  .
Quality factor Q,

67. Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Parallel RLC Circuit
P4.10,
Find i1、 i2、 i v
 i
i1 
 i2

68. ？ Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Parallel RLC Circuit
Review
For sinusoidal circuit， Series ： ？
Parallel :
Two Simple Methods:
Phasor Diagrams and Complex Numbers

69. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Key Words:
Bypass Capacitor
RC Phase Difference
Low-Pass and High-Pass Filter

70. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Bypass Capacitor
P4.11, Let
f ＝ 500Hz，Determine VAB before the C is connected . And VAB after parallel C = 30F
Before C is connected
After C is connected v i

71. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
RC Phase Difference
P4.12,
f = 300Hz, R = 100。 If vo - vi= /4，C =？

72. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Low-Pass and High-Pass Filter
RC---- High-Pass Filter
P4.13, The voltage sources are vi= sin2100t(V), R＝200，
C＝50F， Determine VAC and VDC in output voltage vo.
VDC = 240V

73. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Low-Pass and High-Pass Filter

74. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Low-Pass and High-Pass Filter

75. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis

76. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Low-Pass and High-Pass Filter

77. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis

78. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis

79. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Complex Numbers Analysis
in the circuit of the following fig.
P4.14, Find
v1=120sint v2  i3
 i1
 i2

80. Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Complex Numbers Analysis
P4.15, Let Vm ＝ 100V. Use Thevenin’s theorem to find v v